Approximate Entropy of Brain Network in the Study of Hemispheric Differences

Human brain, a dynamic complex system, can be studied with different approaches, including linear and nonlinear ones. One of the nonlinear approaches widely used in electroencephalographic (EEG) analyses is the entropy, the measurement of disorder in a system. The present study investigates brain networks applying approximate entropy (ApEn) measure for assessing the hemispheric EEG differences; reproducibility and stability of ApEn data across separate recording sessions were evaluated. Twenty healthy adult volunteers were submitted to eyes-closed resting EEG recordings, for 80 recordings. Significant differences in the occipital region, with higher values of entropy in the left hemisphere than in the right one, show that the hemispheres become active with different intensities according to the performed function. Besides, the present methodology proved to be reproducible and stable, when carried out on relatively brief EEG epochs but also at a 1-week distance in a group of 36 subjects. Nonlinear approaches represent an interesting probe to study the dynamics of brain networks. ApEn technique might provide more insight into the pathophysiological processes underlying age-related brain disconnection as well as for monitoring the impact of pharmacological and rehabilitation treatments.     

The prescribed Ricci curvature problem on three‐dimensional unimodular Lie groups

Let G be a three‐dimensional unimodular Lie group, and let T be a left‐invariant symmetric (0,2)‐tensor field on G. We provide the necessary and sufficient conditions on T for the existence of a pair consisting of a left‐invariant Riemannian metric g and a positive constant c such that , where is the Ricci curvature of g. We also discuss the uniqueness of such pairs and show that, in most cases, there exists at most one positive constant c such that is solvable for some left‐invariant Riemannian metric g.     

On Lorentzian Ricci solitons on nilpotent Lie groups

The aim of this article is to exhibit the variety of different Ricci soliton structures that a nilpotent Lie group can support when one allows for the metric tensor to be Lorentzian. In stark contrast to the Riemannian case, we show that a nilpotent Lie group can support a number of non‐isometric Lorentzian Ricci soliton structures with decidedly different qualitative behaviors and that Lorentzian Ricci solitons need not be algebraic Ricci solitons. The analysis is carried out by classifying all left invariant Lorentzian metrics on the connected, simply‐connected five‐dimensional Lie group having a Lie algebra with basis vectors and and non‐trivial bracket relations and , investigating the various curvature properties of the resulting families of metrics, and classifying all Lorentzian Ricci soliton structures.     

Locally symmetric left invariant Riemannian metrics on 3‐dimensional Lie groups

In this paper, we use the powerful tool Milnor bases to determine all the locally symmetric left invariant Riemannian metrics up to automorphism, on 3‐dimensional connected and simply connected Lie groups, by solving system of polynomial equations of constants structure of each Lie algebra . Moreover, we show that E ₀(2) is the only 3‐dimensional Lie group with locally symmetric left invariant Riemannian metrics which are not symmetric.     

The representations of nested composition algebras

In this paper, we investigate the basis graph of the monoid algebra of a submonoid of the monoid of mappings from N = {1,…,n} to itself, defined by a nested sequence of compositions of N. Each such monoid is a left regular band (LRB), that is, a semigroup S satisfying x² = x and xyx = xy for all x,y∈S. This class is su?ciently rich that every path algebra of an acyclic quiver can be embedded in such a monoid algebra. The multiplication in the monoid algebra has a particularly simple quasi-multiplicative form, allowing definition over the integers. Combining this with a formula for Ext-groups for LRBs due to Margolis et al. [6 Margolis, S., Saliola, F., Steinberg, B. (2015). Combinatorial topology and the global dimension of algebras arising in combinatorics. J. Eur. Math Soc. 17(12):3037–3080.[Crossref], [Web of Science ®] , [Google Scholar]], we get a simple criterion for the nested composition algebras to be hereditary.     

The transport dynamics in complex systems governing by anomalous diffusion modelled with Riesz fractional partial differential equations

In this paper, numerical solutions of fractional Fokker–Planck equations with Riesz space fractional derivatives have been developed. Here, the fractional Fokker–Planck equations have been considered in a finite domain. In order to deal with the Riesz fractional derivative operator, shifted Grünwald approximation and fractional centred difference approaches have been used. The explicit finite difference method and Crank–Nicolson implicit method have been applied to obtain the numerical solutions of fractional diffusion equation and fractional Fokker–Planck equations, respectively. Numerical results are presented to demonstrate the accuracy and effectiveness of the proposed numerical solution techniques. Copyright © 2016 John Wiley & Sons, Ltd.     

心电激励下的人体左心室力学响应分析

人体心脏通过电兴奋引起的心肌收缩实现泵血功能 ,而心肌的力学特性高度依赖于肌纤维结构。本文根据肌纤维旋向和复合材料理论以及电生理心脏模型建立了左心室的有限元机械模型 ,仿真研究了左心室在心电兴奋力作用下的力学响应。结果表明左心室的收缩过程是十分复杂的 ,包括轴向和径向的收缩及绕长轴的不同程度的旋转扭曲。从总体趋势上看 ,心尖处变形最为严重。此外 ,仿真结果还表明 ,心壁应力分布不均匀 ,从内壁到外壁有所减少 ;在心尖和心底部应力较大 ,其中内壁心尖处应力最大。这些结果说明了心脏的力学特性与心肌纤维结构、左心室的几何形状以及电兴奋刺激密切相关。作者将本文的仿真结果与医学图像及其它模型进行了对比分析 ,结果表明了该模型用于分析心肌力学功能特性的可行性     

Numerical treatment of solving singular integral equations by using Sinc approximations

In this paper we propose numerical treatment for singular integral equations. The methods are developed by means of the Sinc approximation with smoothing transformations. Such approximation is an effective technique against the singularities of the equations, and achieves exponential convergence. Therefore the methods improve conventional results where only polynomial convergence have been reported. The resulting algebraic system is solved by least squares approximation and leap frog algorithm. Estimation of errors of the approximate solution is presented. Some experimental tests are presented to show the efficient of the proposed methods.